Likelihood function
\[\begin{align*}
\text{NA}_i &\sim \text{Normal}(\mu_{\text{NA},i}, \sigma_{\text{NA}}) \\
\mu_{\text{NA},i} &= \alpha_{\text{NA},\text{id}[i]} + \beta_{\text{NA},\text{id}[i]}\,\text{time}_i \\
\\
\text{PA}_i &\sim \text{Normal}(\mu_{\text{PA},i}, \sigma_{\text{PA}}) \\
\mu_{\text{PA},i} &= \alpha_{\text{PA},\text{id}[i]} + \beta_{\text{PA},\text{id}[i]}\,\text{time}_i
\end{align*}\]
Varying intercepts and slopes:
\[\begin{align*}
\alpha_{\text{NA},\text{id}[i]} &= \alpha_{\text{NA}} + u_{\alpha,\text{NA},\text{id}[i]} \\
\beta_{\text{NA},\text{id}[i]} &= \beta_{\text{NA}} + u_{\beta,\text{NA},\text{id}[i]} \\
\alpha_{\text{PA},\text{id}[i]} &= \alpha_{\text{PA}} + u_{\alpha,\text{PA},\text{id}[i]} \\
\beta_{\text{PA},\text{id}[i]} &= \beta_{\text{PA}} + u_{\beta,\text{PA},\text{id}[i]}
\end{align*}\]
Random Effects: Multivariate Distribution
\[\begin{align*}
\mathbf{u}_{\text{id}[i]} =
\begin{bmatrix}
u_{\alpha,\text{NA},\text{id}[i]} \\
u_{\beta,\text{NA},\text{id}[i]} \\
u_{\alpha,\text{PA},\text{id}[i]} \\
u_{\beta,\text{PA},\text{id}[i]}
\end{bmatrix}
&\sim \text{MVNormal}(\mathbf{0}, \boldsymbol{\Sigma}) \\
\boldsymbol{\Sigma} &= \mathbf{L} \mathbf{R} \mathbf{L} \quad
\\ \text{with} \quad
\mathbf{L} &= \text{diag}(\sigma_{\alpha,\text{NA}}, \sigma_{\beta,\text{NA}}, \sigma_{\alpha,\text{PA}}, \sigma_{\beta,\text{PA}})
\end{align*}\]
Priors
\[\begin{align*}
\alpha_{\text{NA}}, \alpha_{\text{PA}} &\sim \text{Normal}(0, 0.2) \\
\beta_{\text{NA}}, \beta_{\text{PA}} &\sim \text{Normal}(0, 0.5) \\
\sigma_{\text{NA}}, \sigma_{\text{PA}} &\sim \text{Exponential}(1) \\
\sigma_{\alpha,\text{NA}}, \sigma_{\beta,\text{NA}}, \sigma_{\alpha,\text{PA}}, \sigma_{\beta,\text{PA}} &\sim \text{Exponential}(1) \\
\mathbf{R} &\sim \text{LKJcorr}(2)
\end{align*}\]